Roughly speaking, the concept of data depth amounts to assigning an appropriate ordering to the multivariate data, one with respect to a distance that measures how far away a point is from the center of a data cloud. This ordering is determined by a depth function and, among the variety of depth functions that appear in the literature, one of the most widely used is the so-called half-space depth proposed by [1]}. The half-space depth, \(H(y \mid y^n)\) , of a point \(y\)  relative to a data set \(y^n\)  is determined as the smallest fraction of data points contained in a closed half-space with boundary
through \(y\) . It ranges from 0—for the points that lie beyond the convex hull of the data—to its maximum value \(\frac{1}{2}\)  attained at the Tukey median [2]}. Computation of the half-space depth is discussed in [3]} and can be carried out using the depth.halfspace function in ddalpha R package [4]}. Therefore, for multivariate prediction problems, this suggests the following non-conformity measure
\(\psi _i(y^{n+1}) = \tfrac{1}{2} - H(y_i \mid y_{-i}^{n+1}), \quad i \in I_{n+1},\) 
